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Bayesian Estimation of the isotropy violation in the CMB sky

This study presents a Bayesian approach to estimate isotropy violation in the cosmic microwave background (CMB) sky. The method is applicable to polarization maps and utilizes data from the COBE, Planck, and WMAP satellites. The analysis involves calculating the angular power spectrum and the BipoSH coefficients. The Hamiltonian Monte Carlo method is used for sampling the posterior. The results demonstrate the importance of considering isotropy violation in CMB studies.

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Bayesian Estimation of the isotropy violation in the CMB sky

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  1. Bayesian Estimation of the isotropy violation in the CMB sky Santanu Das University of Wisconsin, Madison & Fermi National Accelerator Laboratory

  2. SIToolBox Estimate the isotropy violation in the CMB sky Applicable to the polarization maps

  3. 1965 1990 2003 2013 COBE satellite measured temperature fluctuations in CMB Planckdata release WMAP measured CMB fluctuation Penzias and Wilson discovered cosmic microwave background 

  4. Fluctuation : 60-70 ~ μK Dipole : 3.538 mK Monopole : 2.726 K

  5. WMAP 2003 COBE 1992 Planck 2013 Fluctuation : 60-70 ~ μK

  6. Any fluctuation on a sphere can be expanded in terms of spherical harmonics are Gausisan So we need some statistical measure The angular power spectrum of the CMB sky can be calculated as Statistical Isotropy Fluctuation : 60-70 ~ μK

  7. Theoretically we can calculate CMB power spectrum

  8. Take SI map 2010 Scan using WMAP beam and scan pattern 2011 2013 Reconstract the map 2014 WMAP-7 detected Statistica Isotropy violation in the CMB sky It is proved that SI violation is coming from WMAP scan strategy Das et. al. 2014, Joshi et. al. 2012 Bennet et. al. 2010

  9. Observed CMB sky is actually not SI Weak lensing Doppler anisotropy Non circular Satellite Beam and scan pattern Anisotropic Noise Masking Nonstandard topology of the universe The data analysis has to be proper

  10. What is SI violation If there is any pattern in the sky map then that is SI violated

  11. What is SI violation

  12. What is SI violation

  13. Why do we consider SI violation 1.00 x + 0.05 x 0.95 x 0.95 x + 0.05 x

  14. Why do we consider SI violation BipoSH Hajian & Souradeep, 2003

  15. Why do we consider SI violation 1.00 x + 0.05 x 0.95 x 0.95 x + 0.05 x

  16. Is it sufficient to calculate only BipoSH-es? Expand in spherical harmonics : Bayesian measurement of the BipoSH is important i.e. we need the average and the posterior of the BipoSH Jointly Calculate :

  17. Observed Sky Temp Original Sky Temp Noise Observed sky : Spherical Harmonics basis : We use Hamiltonian Monte Carlo method for sampling the Posterior

  18. What is HMC sampling x2 Takepiand evaluate on time Takepiand evaluate on time Evaluate x1

  19. All these matrices are really big 107 x 107 inverting the matrices are difficult

  20. White Noise Noise matrix is diagonal in pixel space Assume diagonally dominated Expand it in the Taylor series Das 2018, Shaikh et. al. 2018

  21. Analysis and Results SI + 10 μ K SI map + 10 μK Calculate BipoSH by minimizing all BipoSH coefficients upto L=2

  22. Analysis and Results SI + 10 μ K Consistent with 0

  23. Analysis and Results SI + 10 μ K Consistent with 0

  24. Analysis and Results Take SI map Scan using WMAP beam and scan pattern SI + 10 μ K nSI + 10 μ K (Anisotropic) Reconstract the map

  25. Analysis and Results SI + 10 μ K nSI + Scan + Anisotropic noise Noise : 10μK Noise : 30μK

  26. Analysis and Results Take SI map Scan using WMAP beam and scan pattern SI + 10 μ K nSI + 10 μ K (Anisotropic) Reconstract the map nSI + Scan + Anisotropic noise + Mask

  27. Analysis and Results SI + 10 μ K nSI + Scan + Anisotropic noise nSI + Scan + Anisotropic noise + Mask Noise : 30μK

  28. Analysis and Results SI + 10 μ K Mask Dipole modulated map Noise variance Noise Sample SI + Scan + Anisotropic noise ( 1 + x ) Dipole modulation + 10 μ K A recovered sample map Input map to SIToolBox Das 2018

  29. Analysis and Results Doppler Boost Mukherjee et. al., 2015 Generate NSI map using CoNIGS 10 μK noise Mukherjee et. al., 2013

  30. Analysis and Results We can recover all the input values properly

  31. Summary Clis not sufficient to provide full information about the cosmological model in case of Statistical isotropy violation. We need the BipoSH coefficients also. We develop a formalism to calculate Cl and BipoSH coefficients together from a noisy map using complete Bayesian technique. We use Hamiltonian Monte Carlo for sampling the posterior. We apply our algorithm on different simulated maps and recover the BipoSH coefficients appropriately. The codes are made public through github. They are readily applicable to the polarization skymap and other random data over a sphere.

  32. Thank you for being awake Any question?

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